Question

The value of ${\left(\frac{3}{5}\right)}^4{\left(\frac{8}{5}\right)}^{-12}{\left(\frac{32}{5}\right)}^6$ is $\frac{2025}{m}$. Find the value of $m$.

Answer 1

${\left(\frac{3}{5}\right)}^4{\left(\frac{8}{5}\right)}^{-12}{\left(\frac{32}{5}\right)}^6$

$={\left(\frac{3}{5}\right)}^4{\left(\frac{5}{8}\right)}^{12}{\left(\frac{32}{5}\right)}^6$

$=\frac{3^4}{5^4}\times \frac{5^{12}}{8^{12}}\times \frac{{32}^6}{5^6}$

$=\frac{3^4\times 5^{12}\times {\left(2^5\right)}^6}{5^{10}\times {\left(2^3\right)}^{12}}$

$=\frac{3^4\times 5^2\times 2^{30}}{2^{36}}$

$=\frac{3^4\times 5^2}{2^6}$

$=\frac{81\times 25}{64}$

$=\frac{2025}{64}$

Hence, $m=64$.